A convergent series representation for the density of the supremum of a stable process

Abstract

We study the density of the supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ - except for a dense set of Lebesgue measure zero - the asymptotic series which were obtained in Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the supremum.

J.C. Oxtoby.
Measure and category. A survey of the analogies between topological and measure spaces.
Volume 2 of Graduate Texts in
Mathematics.
Springer-Verlag, New York, second edition, 1980.
Math. Review 0584443